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Let M be a 3 × 3 integer matrix each of whose eigenvalues is greater than 1 in modulus and let D ⊂ Z3 be a set with |D| = |det M|, called a digit set. The set equation MT = T + D uniquely defines a nonempty compact set T ⊂ R3. If T has positive Lebesgue measure it is called a 3-dimensional self-affine tile. In the present paper we study topological properties of 3-dimensional self-affine tiles with collinear digit set , i.e., with a digit set of the form D = {0, v, 2v, . . . , (|det M|−1)v} for some v ∈ Z3 \ {0}. We prove that the boundary of such a tile T is homeomorphic to a 2-sphere whenever its set of neighbors in a lattice tiling which is induced by T in a natural way contains 14 elements. The combinatorics of this lattice tiling is then the same as the one of the bitruncated cubic honeycomb , a body-centered cubic lattice tiling by truncated octahedra. We give a characterization of 3-dimensional self-affine tiles with collinear digit set having 14 neighbors in terms of the coefficients of the characteristic polynomial of M. In our proofs we use results of R. H. Bing on the topological characterization of spheres. [ABSTRACT FROM AUTHOR]

Given graphs G and H, a family of vertex-disjoint copies of H in G is called an H-tiling. Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ>0, there exists C>0 such that if p ≥ Cn−1/m2(H), then asymptotically almost surely every spanning subgraph G of the random graph G(n, p) with minimum degree at least δ(G) ≥ (1 − 1/χcr(H) + γ)np contains an H-tiling that covers all but at most γn vertices. Here, χcr(H) denotes the critical chromatic number, a parameter introduced by Komlós, and m2(H) is the 2-density of H. We show that this theorem can be bootstrapped to obtain an H-tiling covering all but at most γ(C/p)m2(H) vertices, which is strictly smaller when p ≥ Cn−1/m2(H). In the case where H = K3, this answers the question of Balogh, Lee and Samotij. Furthermore, for an arbitrary graph H we give an upper bound on p for which some leftover is unavoidable and a bound on the size of a largest H -tiling for p below this value. [ABSTRACT FROM AUTHOR]

We present numerical simulations of electromagnetic millimeter-wave propagation in a two-dimensional lattice of dielectric rods arranged in a tenfold Penrose tiling. We find (i) isotropic photonic band gap as expected for quasicrystals and (ii) localized states. We demonstrate that the high frequency edge of the second band gap is characterized by a very small refractive index (fast light). We study the transmission of electromagnetic waves in the frequency range corresponding to fast light and demonstrate that it is related to tunneling through localized states. We use the fast light phenomenon to design a focusing device—a planoconcave lens. [ABSTRACT FROM AUTHOR]

The pseudoheptagonal mosaic panel above the entrance to the mausoleum of Hodja Ahmad, Shah-i-Zindeh Necropolis in Samarkand, Uzbekistan consists of an intercrossed rib pattern on dark green background. It is a complex combination of eightfold rosettes with heptagons which enclose sevenfold stars. Analysis shows that it is based on a mixture of underlying 3.4.32.4 and 33.42 configurations, invisible in the final pattern. Eightfold elements are regular whereas the sevenfold ones are pseudoheptagonal approximations. Additional, larger-scale circles and festoons which result from the basic geometry, further enliven the pattern. The Hodja Ahmad mosaic may be understood as an attempt to develop further the heptagonal patterns based on the underlying 3.4.32.4 tiling of squares and triangles. [ABSTRACT FROM AUTHOR]

The parallel efficiency is raised by increasing the locality of calculation. With the locally recursive non-locally asynchronous algorithms method, we have constructed a new algorithm that improves the locality of the cross-stencil scheme implementation by the decomposition of the 3D computational domain in time and space. The decomposition is based on a tiling of the 3D1T space into hexahedrons that closely fit the octahedron shape. This shape leads to an algorithm that is less intuitive than the rectangular domain decomposition, but since it follows the natural shape of the dependency region of the cross stencil, it has advantages in data localization and parallelization possibilities. We show its construction, analysis, and implementation possibilities. We present the benchmark results and show that the algorithm follows quantitative estimations: The performance exceeds the memory-bound limit of the stepwise implementation and does not degrade when the whole domain data do not fit higher cache levels. [ABSTRACT FROM AUTHOR]

Given two k -graphs (k-uniform hypergraphs) F and H , a perfect F-tiling (or F-factor) in H is a set of vertex-disjoint copies of F that together cover the vertex set of H. For all complete k-partite k-graphs K, Mycroft proved a minimum codegree condition that guarantees a K-factor in an n-vertex k-graph, which is tight up to an error term o(n). In this paper we improve the error term in Mycroft's result to a sublinear term that relates to the Turán number of K when the differences of the sizes of the vertex classes of K are co-prime. Furthermore, we find a construction which shows that our improved codegree condition is asymptotically tight in infinitely many cases, thus disproving a conjecture of Mycroft. Finally, we determine exact minimum codegree conditions for tiling K(k) (1, . . . , 1, 2) and tiling loose cycles, thus generalizing the results of Czygrinow, DeBiasio and Nagle, and of Czygrinow, respectively. [ABSTRACT FROM AUTHOR]

Induced tilings T = T Kr of the d-dimensional torus 𝕋d generated by an embedded karyon Kr are considered. The operations of differentiation σ : T → T σ are defined; as a result, the induced tilings T σ = T Kr σ of the same torus 𝕋d, generated by the derivative karyon Krσ, are obtained. In terms of karyons Kr, the differentiations σ reduce to a combination of geometric transformations of the space ℝd. It is proved that if the karyon Kr is unimodular, then it generates an induced tiling T = T Kr , and the derivative karyons Krσ are unimodular as well, whence the corresponding derivative tilings T σ = T Kr σ exist. Using unimodular karyons, one can construct an infinite family of induced tilings T = T (α, Kr*), depending on a shift vector α of the torus 𝕋d and an initial karyon Kr*. Two algorithms for constructing such unimodular karyons Kr* are presented. [ABSTRACT FROM AUTHOR]

Given a polygonal substitution tiling T of the plane with subdivision rule τ, we study the conformal tilings Tn associated with τnT. We prove, under the assumption that there is somewhere a non-real similarity mapping one tile to another, that aggregate tiles within Tn converge in shape as n→∞ to their associated Euclidean tiles in T. [ABSTRACT FROM AUTHOR]

It is called a Cairo tiling, because, I had been told, similar tilings can be found in Egypt on the streets of Cairo. Graph: Figure 1This Cairo tiling minimizes the perimeter among tilings by unit-area convex pentagons. [Extracted from the article]

The vertex graph of the Ammann–Beenker tiling is a well‐known quasiperiodic graph with an eightfold rotational symmetry. The coordination sequence and coordination shells of this graph are studied. It is proved that there exists a limit growth form for the vertex graph of the Ammann–Beenker tiling. This growth form is an explicitly calculated regular octagon. Moreover, an asymptotic formula for the coordination numbers of the vertex graph of the Ammann–Beenker tiling is also proved. [ABSTRACT FROM AUTHOR]

The theory of spherical tilings is an interesting and fruitful field, attracting, among other researchers, mathematicians. It is a transverse topic crossing several mathematical areas such as geometry, algebra, topology and number theory, but it is also an object of interest for other scientific fields such as chemistry, physics, art and architecture. Here, we make use of GeoGebra to establish some results, describing a class of monohedral spherical tilings and inferring some conjectures. This will highlight how the use of this software has been crucial for the construction of new knowledge in mathematics with applications in different areas of engineering. [ABSTRACT FROM AUTHOR]

We consider pairs of few-body Ising models where each spin enters a bounded number of interaction terms (bonds) such that each model can be obtained from the dual of the other after freezing k spins on large-degree sites. Such a pair of Ising models can be interpreted as a two-chain complex with k being the rank of the first homology group. Our focus is on the case where k is extensive, that is, scales linearly with the number of bonds n. Flipping any of these additional spins introduces a homologically nontrivial defect (generalized domain wall). In the presence of bond disorder, we prove the existence of a low-temperature weak-disorder region where additional summation over the defects has no effect on the free energy density f(T) in the thermodynamical limit and of a high-temperature region where an extensive homological defect does not affect f(T). We also discuss the convergence of the high- and low-temperature series for the free energy density, prove the analyticity of limiting f(T) at high and low temperatures, and construct inequalities for the critical point(s) where analyticity is lost. As an application, we prove multiplicity of the conventionally defined critical points for Ising models on all { f, d} tilings of the infinite hyperbolic plane, where df/(d + f) > 2. Namely, for these infinite graphs, we show that critical temperatures with free and wired boundary conditions differ, T c (f) < T c (w) . [ABSTRACT FROM AUTHOR]

The Tangent Method of Colomo and Sportiello is applied to the study of the asymptotics of domino tilings of large Aztec rectangles, with some fixed distribution of defects along a boundary. The associated non-intersecting lattice path configurations are made of Schröder paths whose weights involve two parameters γ and q keeping track respectively of one particular type of step and of the area below the paths. We predict the arctic curve for an arbitrary distribution of defects, and illustrate our result with a number of examples involving different classes of boundary defects. [ABSTRACT FROM AUTHOR]

This paper proposes a novel functionality in wavelet-based irregular mesh coding, which is interactive region-of-interest (ROI) support. The proposed approach enables the user to define the arbitrary ROIs at the decoder side and to prioritize and decode these regions at arbitrarily high-granularity levels. In this context, a novel adaptive wavelet transform for irregular meshes is proposed, which enables: 1) varying the resolution across the surface at arbitrarily fine-granularity levels and 2) dynamic tiling, which adapts the tile sizes to the local sampling densities at each resolution level. The proposed tiling approach enables a rate-distortion-optimal distribution of rate across spatial regions. When limiting the highest resolution ROI to the visible regions, the fine granularity of the proposed adaptive wavelet transform reduces the required amount of graphics memory by up to 50%. Furthermore, the required graphics memory for an arbitrary small ROI becomes negligible compared to rendering without ROI support, independent of any tiling decisions. Random access is provided by a novel dynamic tiling approach, which proves to be particularly beneficial for large models of over 106 ~ 107 vertices. The experiments show that the dynamic tiling introduces a limited lossless rate penalty compared to an equivalent codec without ROI support. Additionally, rate savings up to 85% are observed while decoding ROIs of tens of thousands of vertices. [ABSTRACT FROM AUTHOR]

This work is devoted to the study of the 12‐fold rosettes frequently encountered in Islamic art, especially in Moroccan ornamentation. Several types of 12‐fold rosettes are described according to the geometric shape of their petals. They have a large number of variants and offer the possibility of building new, previously unknown rosettes, while respecting the construction method Hasba used by master craftsmen. Artisans developed a technique for combining different types of 12‐fold rosettes to construct infinite periodic tiling belonging to the 17 crystallographic groups. This technique enabled them to diversify the repeated patterns based on 12‐fold rosettes. An analysis of their tiling suggests a method based on elementary geometry to build new patterns with different types of 12‐fold overlapped rosettes and their variants. A procedure based on combination of the distances between two overlapped rosettes is then proposed, which enables generation of new periodic and quasiperiodic patterns. [ABSTRACT FROM AUTHOR]

We describe how we have acquired geometrical models of many periodic tilings of regular polygons from two large collections of images. These models are based on a simplification of the representation recently proposed by us that uses complex numbers. We also describe an algorithm for deciding when two representations give the same tiling, which was used to identify coincidences in these collections. [ABSTRACT FROM AUTHOR]

This paper aims at a reduction of periodicity artefacts during a generation of random heterogeneous material models. The traditional concept of the Periodic Unit Cell is compared with a novel approach of the stochastic Wang tiling. Since modelled structures consist of hard circular/spherical particles in a matrix, the algorithm for placement of inclusions is based on themodifiedmolecular dynamics. We introduce two types ofWang tile boundary conditions to decrease periodicity artefacts. Tested samples for 2D applications form sets of both monodisperse and polydisperse microstructures. The overall volume fractions of these samples are approximately 0.2, 0.4, and 0.6, respectively. The generated sets are analysed both visually and statistically via a two-point probability function. An extension of the stochastic Wang tiling enables to create 3D structures, as well. Therefore, artificial periodicity is also investigated on a 3D sample consisting of spherical particles of identical radii distributed in a continuous phase. [ABSTRACT FROM AUTHOR]

In contrast to the situation with self‐affine tiles, the representation of self‐affine multi‐tiles may not be unique (for a fixed dilation matrix). Let K⊂Rn be an integral self‐affine multi‐tile associated with an n×n integral, expansive matrix B and let K tile Rn by translates of Zn. In this work, we propose a stepwise method to decompose K into measure disjoint pieces Kj satisfying K=⋃Kj in such a way that the collection of sets Kj forms an integral self‐affine collection associated with the matrix B and this with a minimum number of pieces Kj. When used on a given measurable subset K which tiles Rn by translates of Zn, this decomposition terminates after finitely many steps if and only if the set K is an integral self‐affine multi‐tile. Furthermore, we show that the minimal decomposition we provide is unique. [ABSTRACT FROM AUTHOR]

A biperiodic alternating link has an alternating quotient link in the thickened torus. In this paper, we focus on semi‐regular links, a class of biperiodic alternating links whose hyperbolic structure can be immediately determined from a corresponding Euclidean tiling. Consequently, we determine the exact volumes of semi‐regular links. We relate their commensurability and arithmeticity to the corresponding tiling, and assuming a conjecture of Milnor, we show there exist infinitely many pairwise incommensurable semi‐regular links with the same invariant trace field. We show that only two semi‐regular links have totally geodesic checkerboard surfaces; these two links satisfy the Volume Density Conjecture. Finally, we give conditions implying that many additional biperiodic alternating links are hyperbolic and admit a positively oriented, unimodular geometric triangulation. We also provide sharp upper and lower volume bounds for these links. [ABSTRACT FROM AUTHOR]

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This paper reports a 32-unit phase-locked dense heterodyne receiver array at $f_{\mathrm{ RF}}=240$ GHz. To synthesize a large receiving aperture without large sidelobe response, this chip has the following two features. The first feature is the small size of the heterodyne receiver unit, which is only $\lambda _{f_{\mathrm{ RF}}}/4\times \lambda _{f_{\mathrm{ RF}}} /2$. It allows for the integration of two interleaved $4\times 4$ arrays within a 1.2 mm2 die area for concurrent steering of two independent beams. Such unit compactness is enabled by the multi-functionality of the receiver structure, which simultaneously accomplishes local oscillator (LO) generation, inter-unit LO synchronization, input wave coupling, and frequency downconversion. The second feature is the high scalability of the array, which is based on a strongly coupled 2-D LO network. Large array size is realizable simply by tiling more receiver units. With the upscaling of the array, our de-centralized design, contrary to its prior centralized counterparts, offers invariant conversion loss and lower LO phase noise. Meanwhile, the entire LO network is also locked to a 75-MHz reference, facilitating phase-coherent pairing with external sub-terahertz transmitters. A chip prototype using a bulk 65-nm CMOS technology is implemented, with a dc power of 980 mW. Phase locking of the 240-GHz LO is achieved among all 32 units, with a measured phase noise of −84 dBc/Hz (1-MHz offset). The measured sensitivity (BW = 1 kHz) of a single unit is 58 fW. Compared to previous square-law detector arrays of comparable scale and density, this chip provides phase-sensitive detection with $\sim 4300\times $ sensitivity improvement. [ABSTRACT FROM AUTHOR]

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall and Spencer in 2002 [14]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/ n , where a dyadic rectangle is any rectangle that can be written in the form [a2− s, (a + 1)2− s] × [b2− t, (b + 1)2− t] for a, b, s, t ∈ ℤ⩾0. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n4.09), which implies that the mixing time is at most O(n5.09). We complement this by showing that the relaxation time is at least Ω(n1.38), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain. [ABSTRACT FROM AUTHOR]

Well-known techniques for tiled code generation are based on the polyhedral model and affine transformations. An alternative approach to generation of tiled code is to correct original rectangular tiles defined for a loop nest by means of the transitive closure of a dependence graph instead of deriving and applying affine transformations. In this paper, we present results of an analysis of basic features of tiles generated due to correction of original rectangular tiles. We introduce procedures which allow us to recognize such features as target tile type (fixed, varied, parametric), dimensionality, size (the number of statement instances within a tile), and loop nest tileability (the percentage of statement instances that can be tiled with rectangular tiles). We consider differences between those features of tiles generated by means of affine transformations and transitive closure. We also discuss results of experiments with PolyBench benchmarks and show how differences in tiles generated with the examined approach and affine transformations affect serial tiled code performance. [ABSTRACT FROM AUTHOR]

This is the second in a series of papers on conformal tilings. The overriding themes here are local isomorphisms, hierarchical structures, and the conformal "type" problem. Conformal tilings were introduced by the authors in 1997 with a conformally regular pentagonal tiling of the plane. This and even more intricate hierarchical patterns arise when tilings are repeatedly subdivided. Deploying a notion of expansion complexes, we build two-way infinite combinatorial hierarchies and then study the associated conformal tilings. For certain subdivision rules the combinatorial hierarchical properties are faithfully mirrored in their concrete conformal realizations. Examples illustrate the theory throughout the paper. In particular, we study parabolic conformal hierarchies that display periodicities realized by Mobius transformations, motivating higher level hierarchies that will emerge in the next paper of this series. [ABSTRACT FROM AUTHOR]

Stochastic growth processes in dimension (2 + 1) were conjectured by D. Wolf, on the basis of renormalization-group arguments, to fall into two distinct universality classes, according to whether the Hessian H ρ of the speed of growth v (ρ) as a function of the average slope ρ satisfies det H ρ > 0 ("isotropic KPZ class") or det H ρ ≤ 0 ("anisotropic KPZ (AKPZ)" class). The former is characterized by strictly positive growth and roughness exponents, while in the AKPZ class fluctuations are logarithmic in time and space. It is natural to ask (a) if one can exhibit interesting growth models with "smooth" stationary states, i.e., with O(1) fluctuations (instead of logarithmically or power-like growing, as in Wolf's picture) and (b) what new phenomena arise when v (·) is not differentiable, so that H ρ is not defined. The two questions are actually related and here we provide an answer to both, in a specific framework. We define a (2 + 1)-dimensional interface growth process, based on the so-called shuffling algorithm for domino tilings. The stationary, non-reversible measures are translation-invariant Gibbs measures on perfect matchings of Z 2 , with 2-periodic weights. If ρ ≠ 0 , fluctuations are known to grow logarithmically in space and to behave like a two-dimensional GFF. We prove that fluctuations grow at most logarithmically in time and that det H ρ < 0 : the model belongs to the AKPZ class. When ρ = 0 , instead, the stationary state is "smooth", with correlations uniformly bounded in space and time; correspondingly, v (·) is not differentiable at ρ = 0 and we extract the singularity of the eigenvalues of H ρ for ρ ∼ 0 . [ABSTRACT FROM AUTHOR]

X-Shaped bolapolyphiles, comprising a linear polyaromatic core with glycerol groups at each end and two chemically different and incompatible chains, fixed to it at opposite sides, were synthesized and found to self-assemble into honeycomb-type liquid crystalline phases with square symmetry. The polyaromatic π-conjugated rods form the cell walls and the resulting prismatic cells of sub-5 nm size are alternatively filled with perfluorocarbon (RF) and the carbosilane chains (RSi). The resulting structures can be represented as either a two-colour snub-square tiling with triangular and square cells or as a chessboard tiling of squares. [ABSTRACT FROM AUTHOR]

We show that bounded changes to the boundary of a lozenge tilings do not affect the local behaviour inside the domain. As a consequence we prove the existence of a local limit in all domains with planar boundary. The proof does not rely on any exact solvability of the model beyond its links with uniform spanning trees. [ABSTRACT FROM AUTHOR]

We study the limit shape of successive coronas of a tiling, which models the growth of crystals. We define basic terminologies and discuss the existence and uniqueness of corona limits, and then prove that corona limits are completely characterized by directional speeds. As an application, we give another proof that the corona limit of a periodic tiling is a centrally symmetric convex polyhedron [see Zhuravlev (St Petersbg Math J 13(2):201-220, 2002) and Maleev and Shutov (Layer-by-layer growth model for partitions, packings, and graphs, Tranzit-X, Vladimir, 2011)]. [ABSTRACT FROM AUTHOR]

Sparse tiling is a technique to fuse loops that access common data, thus increasing data locality. Unlike traditional loop fusion or blocking, the loops may have different iteration spaces and access shared datasets through indirect memory accesses, such as A[map[i]]—hence the name "sparse." One notable example of such loops arises in discontinuous-Galerkin finite element methods, because of the computation of numerical integrals over different domains (e.g., cells, facets). The major challenge with sparse tiling is implementation—not only is it cumbersome to understand and synthesize, but it is also onerous to maintain and generalize, as it requires a complete rewrite of the bulk of the numerical computation. In this article, we propose an approach to extend the applicability of sparse tiling based on raising the level of abstraction. Through a sequence of compiler passes, the mathematical specification of a problem is progressively lowered, and eventually sparse-tiled C for-loops are generated. Besides automation, we advance the state-of-the-art by introducing a revisited, more efficient sparse tiling algorithm; support for distributed-memory parallelism; a range of fine-grained optimizations for increased runtime performance; implementation in a publicly available library, SLOPE; and an in-depth study of the performance impact in Seigen, a real-world elastic wave equation solver for seismological problems, which shows speed-ups up to 1.28× on a platform consisting of 896 Intel Broadwell cores. [ABSTRACT FROM AUTHOR]

Supernode transformation, or tiling, is a technique that partitions algorithms to improve data locality and parallelism to achieve shortest running time. It groups multiple iterations of nested loops into supernodes to be assigned to processors for processing in parallel. A supernode transformation can be described by supernode size and shape. This paper focuses on supernode transformation on General Purpose Graphic Processing Units (GPGPUs), including supernode scheduling, supernode mapping to GPGPU blocks, and the finding of the optimal supernode size, for achieving the shortest total running time. The algorithms considered are two nested loops with regular data dependencies. The Longest Common Subsequence problem is used as an illustration. A novel mathematical model for the total running time is established as a function of the supernode size, algorithm parameters such as the problem size and the data dependence, the computation time of each loop iteration, architecture parameters such as the number of GPGPU blocks, and the communication cost. The optimal supernode size is derived from this closed form model. The model and the optimal supernode size provide better results than previous research and are verified by simulations on GPGPUs. Iterations in a two-dimensional uniform dependence algorithm iteration space of , shown as the intersections in the picture, can be grouped into rectangles of known as a tile or a supernode. This process is called supernode transformation or tiling. It reduces the inter-iteration communication cost thus improves the total execution time. The supernodes on the same wavefront may be scheduled on GPU to be processed at the same time, each by a GPU block. The size of the tile, , plays an important role in this transformation. The optimal size can lead to minimal total execution time. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the halting problem for deterministic cellular automata on the pentagrid. We prove that the problem is decidable when the cellular automaton starts its computation from a finite configuration and when it has two states, one of them being a quiescent state. [ABSTRACT FROM AUTHOR]

A perfect H-tiling in a graph G is a collection of vertex-disjoint copies of a graph H in G that together cover all the vertices in G. In this paper we investigate perfect H-tilings in a random graph model introduced by Bohman, Frieze and Martin [6] in which one starts with a dense graph and then adds m random edges to it. Specifically, for any fixed graph H, we determine the number of random edges required to add to an arbitrary graph of linear minimum degree in order to ensure the resulting graph contains a perfect H-tiling with high probability. Our proof utilizes Szemerédi's Regularity Lemma [29] as well as a special case of a result of Komlós [18] concerning almost perfect H-tilings in dense graphs. [ABSTRACT FROM AUTHOR]

A three-dimensional pentagon-shaped stereo-tiling concept has been realized through use of penta-graphene carbon structures, although mathematically, regular planar pentagon shapes cannot be used to completely tile the Euclidean plane. Two applications of the findings of this study have been considered from the mathematical and engineering viewpoints. First, the proposed discovery facilitates math-rule-based generation of beautiful designs comprising star shapes formed using regular pentagons. The underlying mathematical logic and hexagon-division rules have been deduced to obtain the proposed pentagonal stereo-tiling pattern comprising equal-length bonds, and two aesthetic designs have been derived using the deduced logic. A fence-like structure exclusively comprising pentagram stars has been designed and given the name "Star Walls." Emphasis has also been laid on application of the proposed stereo-tiling concept in industrial design operations, such as emboss manufacturing. To this end, finite element analysis of embossed steel sheets has been performed to verify the feasibility of the said industrial application. [ABSTRACT FROM AUTHOR]

A fractal tiling or f -tiling is a tiling which possesses self-similarity and the boundary of which is a fractal. By substitution rule of tilings, this short paper presents a very simple strategy to create a great number of f -tilings. The substitution tiling Equithirds is demonstrated to show how to achieve it in detail. The method can be generalized to every tiling that can be constructed by substitution rule. [ABSTRACT FROM AUTHOR]

Digital mosaics have usually used regular tiles, simulating historical tessellated mosaics. In this paper, we present a method for synthesizing pebble mosaics, a historical mosaic style in which the tiles are rounded pebbles. We address both the tiling problem, of distributing pebbles over the image plane so as to approximate the input image content, and the problem of geometry, creating a smooth rounded shape for each pebble. We adopt simple linear iterative clustering (SLIC) to obtain elongated tiles conforming to image content, and smooth the resulting irregular shapes into shapes resembling pebble cross-sections. Then, we create an interior and exterior contour for each pebble and solve a Laplace equation over the region between them to obtain height-field geometry. The resulting pebble set approximates the input image while representing full geometry that can be rendered and textured for a highly detailed representation of a pebble mosaic. [ABSTRACT FROM AUTHOR]

Mobile devices have seen their performance increased in latest years due to improvements on System on Chip technologies. These shared memory systems now integrate multicore CPUs and accelerators, and obtaining the optimal performance from such heterogeneous architectures requires making use of accelerators in an efficient way. Graphics Processing Units (GPUs) are accelerators that often outperform multicore CPUs in data-parallel workloads by orders of magnitude, so their use for image processing applications on mobile devices is very important. In this work we explore tiling code optimizations for GPU applications running on mobile devices. A dynamic adaptive tile size selection methodology is created, which allows finding at runtime close-to-optimal parameterizations independently of the underlying architecture. Results demonstrate the performance benefits of these optimizations over a set of stencil-based image processing benchmarks. [ABSTRACT FROM AUTHOR]

Based on the recent development of sampling and reconstruction results for slowly time-varying single-input single-output channel operators, we derive sampling results in the multiple-input multiple-output setting where all subchannels satisfy an underspread condition, that is, their spreading functions are supported on individual sets of small measure. At the center of our work is the extension of the single-input single-output dual tiling condition to this setting; it characterizes which periodic weighted delta trains can be used to identify a given class of multiple-input multiple-output channel operators satisfying a spreading support constraint. Building on the dual tiling condition, we compute reconstruction formulas for the operator's symbol in closed form and discuss the problem of identifying multiple-input multiple-output operators where only restrictions in size, but not on location and geometry, of the subchannel spreading supports are known. [ABSTRACT FROM AUTHOR]

We prove that for any infinite countable amenable group G, any ε > 0 {\varepsilon>0} and any finite subset K ⊂ G {K\subset G} , there exists a tiling (partition of G into finite "tiles" using only finitely many "shapes"), where all the tiles are (K , ε) {(K,\varepsilon)} -invariant. Moreover, our tiling has topological entropy zero (i.e., subexponential complexity of patterns). As an application, we construct a free action of G (in the sense that the mappings, associated to elements of G other than the unit, have no fixed points) on a zero-dimensional space, such that the topological entropy of this action is zero. [ABSTRACT FROM AUTHOR]

The article offers information on chapter 23 of the book "The Plaid Model," by Richard Evan Schwartz. It explores several proofs for the theorem regarding plaid PET and discusses the use of Box Theorem and the Copy Theorem for approximation of sequence of even rational. It also discusses proofs for the Box and Copy Theorem and hidden symmetries between plaid model and Truchet tiling.

The article presents the chapter eight of the book "The Plaid Model," by Richard Evan Schwartz. Topics include the proof of the Plaid Master Picture Theorem, the generation of union PLA of plaid polygons through explicitly defined tiling classifying space, and the partition of space X into convex integer polytopes.

This work considers the scaling properties characterizing the hyperuniformity (or anti‐hyperuniformity) of long‐wavelength fluctuations in a broad class of one‐dimensional substitution tilings. A simple argument is presented which predicts the exponent α governing the scaling of Fourier intensities at small wavenumbers, tilings with α > 0 being hyperuniform, and numerical computations confirm that the predictions are accurate for quasiperiodic tilings, tilings with singular continuous spectra and limit‐periodic tilings. Quasiperiodic or singular continuous cases can be constructed with α arbitrarily close to any given value between −1 and 3. Limit‐periodic tilings can be constructed with α between −1 and 1 or with Fourier intensities that approach zero faster than any power law. This work examines the long‐wavelength scaling properties of self‐similar substitution tilings, placing them in their hyperuniformity classes. Quasiperiodic, non‐PV (Pisot–Vijayaraghavan number) and limit‐periodic examples are analyzed. Novel behavior is demonstrated for certain limit‐periodic cases. [ABSTRACT FROM AUTHOR]

An elementary method is described for finding the coordination sequences for a tiling, based on coloring the underlying graph. The first application is to the two kinds of vertices (tetravalent and trivalent) in the Cairo (or dual‐32.4.3.4) tiling. The coordination sequence for a tetravalent vertex turns out, surprisingly, to be 1, 4, 8, 12, 16, ..., the same as for a vertex in the familiar square (or 44) tiling. The authors thought that such a simple fact should have a simple proof, and this article is the result. The method is also used to obtain coordination sequences for the 32.4.3.4, 3.4.6.4, 4.82, 3.122 and 34.6 uniform tilings, and the snub‐632 tiling. In several cases the results provide proofs for previously conjectured formulas. This article presents a simple method for finding formulas for coordination sequences, based on coloring the underlying graph according to certain rules. It is illustrated by applying it to several uniform tilings and their duals. [ABSTRACT FROM AUTHOR]

In this article, a framework is presented that allows the systematic derivation of planar edge‐to‐edge k‐isocoronal tilings from tile‐s‐transitive tilings, s ≤ k. A tiling is k‐isocoronal if its vertex coronae form k orbits or k transitivity classes under the action of its symmetry group. The vertex corona of a vertex x of is used to refer to the tiles that are incident to x. The k‐isocoronal tilings include the vertex‐k‐transitive tilings (k‐isogonal) and k‐uniform tilings. In a vertex‐k‐transitive tiling, the vertices form k transitivity classes under its symmetry group. If this tiling consists of regular polygons then it is k‐uniform. This article also presents the classification of isocoronal tilings in the Euclidean plane. This article presents a method to determine planar edge‐to‐edge k‐isocoronal tilings – tilings whose vertex coronae form k orbits or k transitivity classes under the action of the symmetry group. [ABSTRACT FROM AUTHOR]

Komlós [Komlós: Tiling Turán Theorems, Combinatorica, 2000] determined the asymptotically optimal minimum‐degree condition for covering a given proportion of vertices of a host graph by vertex‐disjoint copies of a fixed graph H, thus essentially extending the Hajnal–Szemerédi theorem that deals with the case when H is a clique. We give a proof of a graphon version of Komlós's theorem. To prove this graphon version, and also to deduce from it the original statement about finite graphs, we use the machinery introduced in [Hladký, Hu, Piguet: Tilings in graphons, arXiv:1606.03113]. We further prove a stability version of Komlós's theorem. [ABSTRACT FROM AUTHOR]

The linoleum was more than a prop for talks; it gave John his gut-level understanding of the tiling's properties. John's names, apt, short, and compelling, became tools for studying Penrose tiles and tilings and a language for discussing them. After one of John's talks on Penrose tiles, a smitten member of the audience came up to him and gushed, "Oh, Mr. Conway! Soon after Martin Gardner rolled out the Penrose tilings in I Scientific American i ,[4] John drew a large portion of a Penrose tiling on a sheet of linoleum and rolled it out on the stage. [Extracted from the article]